Abstract

Many experiments have been made to test the Clauser–Horne–Shimony–Holt (CHSH)–Bell inequality (CBI). However, the Cirel’son’s limit has not been reached yet, due to various practically existing imperfections. In this paper, we experimentally investigate how to select the measurement settings for obtaining the optimal violation of CBI for a mixed entangled state. With the commercialized entangled source, we first determine the entanglement degree of the generated photon pairs by using the tomographic technique, and then numerically optimize the measurement settings. Finally, we experimentally obtain the optimal violation of the CBI with the present commercialized entangled source without the maximal entanglement.

© 2012 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  5. P. G. Kwiat, K. Mattle, H. Weinfurter, and A. Zeilinger, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
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  6. M. A. Rowe, D. Kielpinski, V. Meyer, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland, “Experimental violation of a Bell’s inequality with efficient detection,” Nature 409, 791–794 (2001).
    [CrossRef]
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    [CrossRef]
  14. D. Magde and H. Mahr, “Study in ammonium dihydrogen phosphate of spontaneous parametric interaction tunable from 4400 to 16 000 A,” Phys. Rev. Lett. 18, 905–907(1967).
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  15. F. Verstraete and M. M. Wolf, “Entanglement versus Bell violations and their behavior under local filtering operations,” Phys. Rev. Lett. 89, 170401 (2002).
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    [CrossRef]
  20. S. L. Braunstein, A. Mann, and M. Revzen, “Maximal violation of Bell inequalities for mixed states,” Phys. Rev. Lett. 68, 3259–3261 (1992).
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  21. R. Horodecki, P. Horodecki, and M. Horodecki, “Violating Bell inequality by mixed spin-1/2 states: necessary and sufficient condition,” Phys. Lett. A 200, 340–344 (1995).
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    [CrossRef]
  25. M. A. Nielsen and I. L. Chuang, Quantum Computation and Information (Cambridge University, 2000).
  26. V. Coffman, J. Kundu, and W. K. Wootters, “Distributed entanglement,” Phys. Rev. A 61, 052306 (2000).
    [CrossRef]
  27. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd ed. (Cambridge University, 1992).
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2011 (1)

E. Pomarico, J. D. Bancal, B. Sanguinetti, A. Rochdi, and N. Gisin, “Various quantum nonlocality tests with a commercial two-photon entanglement source,” Phys. Rev. A 83, 052104 (2011).
[CrossRef]

2010 (1)

G. Lima, G. Vallone, A. Chiuri, A. Cabello, and P. Mataloni, “Experimental Bell-inequality violation without the postselection loophole,” Phys. Rev. A 81, 040101(R) (2010).
[CrossRef]

2009 (1)

M. Ansmann, H. Wang, R. C. Bialczak, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, A. N. Cleland, and J. M. Martinis, “Violation of Bell’s inequality in Josephson phase qubits,” Nature 461, 504–506 (2009).
[CrossRef]

2006 (2)

L. Wei, Y. Liu, M. Storcz, and F. Nori, “Macroscopic Einstein-Podolsky-Rosen pairs in superconducting circuits,” Phys. Rev. A 73, 052307 (2006).
[CrossRef]

H. Sakai, T. Saito, T. Ikeda, K. Itoh, T. Kawabata, H. Kuboki, Y. Maeda, N. Matsui, C. Rangacharyulu, M. Sasano, Y. Satou, K. Sekiguchi, K. Suda, A. Tamii, T. Uesaka, and K. Yako, “Spin correlations of strongly interacting massive fermion pairs as a test of Bell’s inequality,” Phys. Rev. Lett. 97, 150405 (2006).
[CrossRef]

2005 (1)

L. Wei, Y. Liu, and F. Nori, “Testing Bells inequality in a constantly coupled Josephson circuit by effective single-qubit operations,” Phys. Rev. B 72, 104516 (2005).
[CrossRef]

2002 (2)

F. Verstraete and M. M. Wolf, “Entanglement versus Bell violations and their behavior under local filtering operations,” Phys. Rev. Lett. 89, 170401 (2002).
[CrossRef]

D. Dehlinger and M. W. Mitchell, “Entangled photons, nonlocality, and Bell inequalities in the undergraduate laboratory,” Am. J. Phys. 70, 903–910 (2002).
[CrossRef]

2001 (3)

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[CrossRef]

Y. F. Huang, C. F. Li, Y. S. Zhang, and G. C. Guo, “Experimental test of CHSH inequality for non-maximally entangled states,” Phys. Lett. Sect. A 287, 317–320 (2001).
[CrossRef]

M. A. Rowe, D. Kielpinski, V. Meyer, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland, “Experimental violation of a Bell’s inequality with efficient detection,” Nature 409, 791–794 (2001).
[CrossRef]

2000 (1)

V. Coffman, J. Kundu, and W. K. Wootters, “Distributed entanglement,” Phys. Rev. A 61, 052306 (2000).
[CrossRef]

1998 (1)

G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, “Violation of Bell’s inequality under strict Einstein locality conditions,” Phys. Rev. Lett. 81, 5039–5043(1998).
[CrossRef]

1995 (2)

R. Horodecki, P. Horodecki, and M. Horodecki, “Violating Bell inequality by mixed spin-1/2 states: necessary and sufficient condition,” Phys. Lett. A 200, 340–344 (1995).
[CrossRef]

P. G. Kwiat, K. Mattle, H. Weinfurter, and A. Zeilinger, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[CrossRef]

1992 (1)

S. L. Braunstein, A. Mann, and M. Revzen, “Maximal violation of Bell inequalities for mixed states,” Phys. Rev. Lett. 68, 3259–3261 (1992).
[CrossRef]

1989 (1)

R. F. Werner, “Quantum states with Einstein Podolsky Rosen correlations admitting a hidden-variable model,” Phys. Rev. A 40, 4277–4281 (1989).
[CrossRef]

1981 (1)

A. Aspect, P. Grangier, and G. Roger, “Experimental tests of realistic local theories via Bell’s theorem,” Phys. Rev. Lett. 47, 460–463 (1981).
[CrossRef]

1980 (1)

B. S. Cirel’son, “Quantum generalizations of Bell’s inequality,” Lett. Math. Phys. 4, 93–100 (1980).
[CrossRef]

1969 (1)

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
[CrossRef]

1967 (1)

D. Magde and H. Mahr, “Study in ammonium dihydrogen phosphate of spontaneous parametric interaction tunable from 4400 to 16 000 A,” Phys. Rev. Lett. 18, 905–907(1967).
[CrossRef]

1964 (1)

J. Bell, “On the Einstein Podolsky Rosen paradox,” Physics 1, 195–200 (1964).

1935 (1)

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?,” Phys. Rev. 47, 777–780 (1935).
[CrossRef]

Ansmann, M.

M. Ansmann, H. Wang, R. C. Bialczak, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, A. N. Cleland, and J. M. Martinis, “Violation of Bell’s inequality in Josephson phase qubits,” Nature 461, 504–506 (2009).
[CrossRef]

Aspect, A.

A. Aspect, P. Grangier, and G. Roger, “Experimental tests of realistic local theories via Bell’s theorem,” Phys. Rev. Lett. 47, 460–463 (1981).
[CrossRef]

Bancal, J. D.

E. Pomarico, J. D. Bancal, B. Sanguinetti, A. Rochdi, and N. Gisin, “Various quantum nonlocality tests with a commercial two-photon entanglement source,” Phys. Rev. A 83, 052104 (2011).
[CrossRef]

Bell, J.

J. Bell, “On the Einstein Podolsky Rosen paradox,” Physics 1, 195–200 (1964).

Bialczak, R. C.

M. Ansmann, H. Wang, R. C. Bialczak, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, A. N. Cleland, and J. M. Martinis, “Violation of Bell’s inequality in Josephson phase qubits,” Nature 461, 504–506 (2009).
[CrossRef]

Braunstein, S. L.

S. L. Braunstein, A. Mann, and M. Revzen, “Maximal violation of Bell inequalities for mixed states,” Phys. Rev. Lett. 68, 3259–3261 (1992).
[CrossRef]

Cabello, A.

G. Lima, G. Vallone, A. Chiuri, A. Cabello, and P. Mataloni, “Experimental Bell-inequality violation without the postselection loophole,” Phys. Rev. A 81, 040101(R) (2010).
[CrossRef]

Chiuri, A.

G. Lima, G. Vallone, A. Chiuri, A. Cabello, and P. Mataloni, “Experimental Bell-inequality violation without the postselection loophole,” Phys. Rev. A 81, 040101(R) (2010).
[CrossRef]

Chuang, I. L.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Information (Cambridge University, 2000).

Cirel’son, B. S.

B. S. Cirel’son, “Quantum generalizations of Bell’s inequality,” Lett. Math. Phys. 4, 93–100 (1980).
[CrossRef]

Clauser, J. F.

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
[CrossRef]

Cleland, A. N.

M. Ansmann, H. Wang, R. C. Bialczak, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, A. N. Cleland, and J. M. Martinis, “Violation of Bell’s inequality in Josephson phase qubits,” Nature 461, 504–506 (2009).
[CrossRef]

Coffman, V.

V. Coffman, J. Kundu, and W. K. Wootters, “Distributed entanglement,” Phys. Rev. A 61, 052306 (2000).
[CrossRef]

Dehlinger, D.

D. Dehlinger and M. W. Mitchell, “Entangled photons, nonlocality, and Bell inequalities in the undergraduate laboratory,” Am. J. Phys. 70, 903–910 (2002).
[CrossRef]

Einstein, A.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?,” Phys. Rev. 47, 777–780 (1935).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd ed. (Cambridge University, 1992).

Gisin, N.

E. Pomarico, J. D. Bancal, B. Sanguinetti, A. Rochdi, and N. Gisin, “Various quantum nonlocality tests with a commercial two-photon entanglement source,” Phys. Rev. A 83, 052104 (2011).
[CrossRef]

Grangier, P.

A. Aspect, P. Grangier, and G. Roger, “Experimental tests of realistic local theories via Bell’s theorem,” Phys. Rev. Lett. 47, 460–463 (1981).
[CrossRef]

Guo, G. C.

Y. F. Huang, C. F. Li, Y. S. Zhang, and G. C. Guo, “Experimental test of CHSH inequality for non-maximally entangled states,” Phys. Lett. Sect. A 287, 317–320 (2001).
[CrossRef]

Hofheinz, M.

M. Ansmann, H. Wang, R. C. Bialczak, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, A. N. Cleland, and J. M. Martinis, “Violation of Bell’s inequality in Josephson phase qubits,” Nature 461, 504–506 (2009).
[CrossRef]

Holt, R. A.

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
[CrossRef]

Horne, M. A.

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
[CrossRef]

Horodecki, M.

R. Horodecki, P. Horodecki, and M. Horodecki, “Violating Bell inequality by mixed spin-1/2 states: necessary and sufficient condition,” Phys. Lett. A 200, 340–344 (1995).
[CrossRef]

Horodecki, P.

R. Horodecki, P. Horodecki, and M. Horodecki, “Violating Bell inequality by mixed spin-1/2 states: necessary and sufficient condition,” Phys. Lett. A 200, 340–344 (1995).
[CrossRef]

Horodecki, R.

R. Horodecki, P. Horodecki, and M. Horodecki, “Violating Bell inequality by mixed spin-1/2 states: necessary and sufficient condition,” Phys. Lett. A 200, 340–344 (1995).
[CrossRef]

Huang, Y. F.

Y. F. Huang, C. F. Li, Y. S. Zhang, and G. C. Guo, “Experimental test of CHSH inequality for non-maximally entangled states,” Phys. Lett. Sect. A 287, 317–320 (2001).
[CrossRef]

Ikeda, T.

H. Sakai, T. Saito, T. Ikeda, K. Itoh, T. Kawabata, H. Kuboki, Y. Maeda, N. Matsui, C. Rangacharyulu, M. Sasano, Y. Satou, K. Sekiguchi, K. Suda, A. Tamii, T. Uesaka, and K. Yako, “Spin correlations of strongly interacting massive fermion pairs as a test of Bell’s inequality,” Phys. Rev. Lett. 97, 150405 (2006).
[CrossRef]

Itano, W. M.

M. A. Rowe, D. Kielpinski, V. Meyer, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland, “Experimental violation of a Bell’s inequality with efficient detection,” Nature 409, 791–794 (2001).
[CrossRef]

Itoh, K.

H. Sakai, T. Saito, T. Ikeda, K. Itoh, T. Kawabata, H. Kuboki, Y. Maeda, N. Matsui, C. Rangacharyulu, M. Sasano, Y. Satou, K. Sekiguchi, K. Suda, A. Tamii, T. Uesaka, and K. Yako, “Spin correlations of strongly interacting massive fermion pairs as a test of Bell’s inequality,” Phys. Rev. Lett. 97, 150405 (2006).
[CrossRef]

James, D. F. V.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[CrossRef]

Jennewein, T.

G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, “Violation of Bell’s inequality under strict Einstein locality conditions,” Phys. Rev. Lett. 81, 5039–5043(1998).
[CrossRef]

Kawabata, T.

H. Sakai, T. Saito, T. Ikeda, K. Itoh, T. Kawabata, H. Kuboki, Y. Maeda, N. Matsui, C. Rangacharyulu, M. Sasano, Y. Satou, K. Sekiguchi, K. Suda, A. Tamii, T. Uesaka, and K. Yako, “Spin correlations of strongly interacting massive fermion pairs as a test of Bell’s inequality,” Phys. Rev. Lett. 97, 150405 (2006).
[CrossRef]

Kielpinski, D.

M. A. Rowe, D. Kielpinski, V. Meyer, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland, “Experimental violation of a Bell’s inequality with efficient detection,” Nature 409, 791–794 (2001).
[CrossRef]

Knight, P. L.

P. L. Knight and A. Miller, Measuring the Quantum States of Light (Cambridge University, 1997).

Kuboki, H.

H. Sakai, T. Saito, T. Ikeda, K. Itoh, T. Kawabata, H. Kuboki, Y. Maeda, N. Matsui, C. Rangacharyulu, M. Sasano, Y. Satou, K. Sekiguchi, K. Suda, A. Tamii, T. Uesaka, and K. Yako, “Spin correlations of strongly interacting massive fermion pairs as a test of Bell’s inequality,” Phys. Rev. Lett. 97, 150405 (2006).
[CrossRef]

Kundu, J.

V. Coffman, J. Kundu, and W. K. Wootters, “Distributed entanglement,” Phys. Rev. A 61, 052306 (2000).
[CrossRef]

Kwiat, P. G.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[CrossRef]

P. G. Kwiat, K. Mattle, H. Weinfurter, and A. Zeilinger, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[CrossRef]

Li, C. F.

Y. F. Huang, C. F. Li, Y. S. Zhang, and G. C. Guo, “Experimental test of CHSH inequality for non-maximally entangled states,” Phys. Lett. Sect. A 287, 317–320 (2001).
[CrossRef]

Lima, G.

G. Lima, G. Vallone, A. Chiuri, A. Cabello, and P. Mataloni, “Experimental Bell-inequality violation without the postselection loophole,” Phys. Rev. A 81, 040101(R) (2010).
[CrossRef]

Liu, Y.

L. Wei, Y. Liu, M. Storcz, and F. Nori, “Macroscopic Einstein-Podolsky-Rosen pairs in superconducting circuits,” Phys. Rev. A 73, 052307 (2006).
[CrossRef]

L. Wei, Y. Liu, and F. Nori, “Testing Bells inequality in a constantly coupled Josephson circuit by effective single-qubit operations,” Phys. Rev. B 72, 104516 (2005).
[CrossRef]

Lucero, E.

M. Ansmann, H. Wang, R. C. Bialczak, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, A. N. Cleland, and J. M. Martinis, “Violation of Bell’s inequality in Josephson phase qubits,” Nature 461, 504–506 (2009).
[CrossRef]

Maeda, Y.

H. Sakai, T. Saito, T. Ikeda, K. Itoh, T. Kawabata, H. Kuboki, Y. Maeda, N. Matsui, C. Rangacharyulu, M. Sasano, Y. Satou, K. Sekiguchi, K. Suda, A. Tamii, T. Uesaka, and K. Yako, “Spin correlations of strongly interacting massive fermion pairs as a test of Bell’s inequality,” Phys. Rev. Lett. 97, 150405 (2006).
[CrossRef]

Magde, D.

D. Magde and H. Mahr, “Study in ammonium dihydrogen phosphate of spontaneous parametric interaction tunable from 4400 to 16 000 A,” Phys. Rev. Lett. 18, 905–907(1967).
[CrossRef]

Mahr, H.

D. Magde and H. Mahr, “Study in ammonium dihydrogen phosphate of spontaneous parametric interaction tunable from 4400 to 16 000 A,” Phys. Rev. Lett. 18, 905–907(1967).
[CrossRef]

Mann, A.

S. L. Braunstein, A. Mann, and M. Revzen, “Maximal violation of Bell inequalities for mixed states,” Phys. Rev. Lett. 68, 3259–3261 (1992).
[CrossRef]

Martinis, J. M.

M. Ansmann, H. Wang, R. C. Bialczak, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, A. N. Cleland, and J. M. Martinis, “Violation of Bell’s inequality in Josephson phase qubits,” Nature 461, 504–506 (2009).
[CrossRef]

Mataloni, P.

G. Lima, G. Vallone, A. Chiuri, A. Cabello, and P. Mataloni, “Experimental Bell-inequality violation without the postselection loophole,” Phys. Rev. A 81, 040101(R) (2010).
[CrossRef]

Matsui, N.

H. Sakai, T. Saito, T. Ikeda, K. Itoh, T. Kawabata, H. Kuboki, Y. Maeda, N. Matsui, C. Rangacharyulu, M. Sasano, Y. Satou, K. Sekiguchi, K. Suda, A. Tamii, T. Uesaka, and K. Yako, “Spin correlations of strongly interacting massive fermion pairs as a test of Bell’s inequality,” Phys. Rev. Lett. 97, 150405 (2006).
[CrossRef]

Mattle, K.

P. G. Kwiat, K. Mattle, H. Weinfurter, and A. Zeilinger, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[CrossRef]

Meyer, V.

M. A. Rowe, D. Kielpinski, V. Meyer, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland, “Experimental violation of a Bell’s inequality with efficient detection,” Nature 409, 791–794 (2001).
[CrossRef]

Milburn, G. J.

D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, 1994).

Miller, A.

P. L. Knight and A. Miller, Measuring the Quantum States of Light (Cambridge University, 1997).

Mitchell, M. W.

D. Dehlinger and M. W. Mitchell, “Entangled photons, nonlocality, and Bell inequalities in the undergraduate laboratory,” Am. J. Phys. 70, 903–910 (2002).
[CrossRef]

Monroe, C.

M. A. Rowe, D. Kielpinski, V. Meyer, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland, “Experimental violation of a Bell’s inequality with efficient detection,” Nature 409, 791–794 (2001).
[CrossRef]

Munro, W. J.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[CrossRef]

Neeley, M.

M. Ansmann, H. Wang, R. C. Bialczak, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, A. N. Cleland, and J. M. Martinis, “Violation of Bell’s inequality in Josephson phase qubits,” Nature 461, 504–506 (2009).
[CrossRef]

Nielsen, M. A.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Information (Cambridge University, 2000).

Nori, F.

L. Wei, Y. Liu, M. Storcz, and F. Nori, “Macroscopic Einstein-Podolsky-Rosen pairs in superconducting circuits,” Phys. Rev. A 73, 052307 (2006).
[CrossRef]

L. Wei, Y. Liu, and F. Nori, “Testing Bells inequality in a constantly coupled Josephson circuit by effective single-qubit operations,” Phys. Rev. B 72, 104516 (2005).
[CrossRef]

O’Connell, A. D.

M. Ansmann, H. Wang, R. C. Bialczak, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, A. N. Cleland, and J. M. Martinis, “Violation of Bell’s inequality in Josephson phase qubits,” Nature 461, 504–506 (2009).
[CrossRef]

Podolsky, B.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?,” Phys. Rev. 47, 777–780 (1935).
[CrossRef]

Pomarico, E.

E. Pomarico, J. D. Bancal, B. Sanguinetti, A. Rochdi, and N. Gisin, “Various quantum nonlocality tests with a commercial two-photon entanglement source,” Phys. Rev. A 83, 052104 (2011).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd ed. (Cambridge University, 1992).

Rangacharyulu, C.

H. Sakai, T. Saito, T. Ikeda, K. Itoh, T. Kawabata, H. Kuboki, Y. Maeda, N. Matsui, C. Rangacharyulu, M. Sasano, Y. Satou, K. Sekiguchi, K. Suda, A. Tamii, T. Uesaka, and K. Yako, “Spin correlations of strongly interacting massive fermion pairs as a test of Bell’s inequality,” Phys. Rev. Lett. 97, 150405 (2006).
[CrossRef]

Revzen, M.

S. L. Braunstein, A. Mann, and M. Revzen, “Maximal violation of Bell inequalities for mixed states,” Phys. Rev. Lett. 68, 3259–3261 (1992).
[CrossRef]

Rochdi, A.

E. Pomarico, J. D. Bancal, B. Sanguinetti, A. Rochdi, and N. Gisin, “Various quantum nonlocality tests with a commercial two-photon entanglement source,” Phys. Rev. A 83, 052104 (2011).
[CrossRef]

Roger, G.

A. Aspect, P. Grangier, and G. Roger, “Experimental tests of realistic local theories via Bell’s theorem,” Phys. Rev. Lett. 47, 460–463 (1981).
[CrossRef]

Rosen, N.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?,” Phys. Rev. 47, 777–780 (1935).
[CrossRef]

Rowe, M. A.

M. A. Rowe, D. Kielpinski, V. Meyer, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland, “Experimental violation of a Bell’s inequality with efficient detection,” Nature 409, 791–794 (2001).
[CrossRef]

Sackett, C. A.

M. A. Rowe, D. Kielpinski, V. Meyer, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland, “Experimental violation of a Bell’s inequality with efficient detection,” Nature 409, 791–794 (2001).
[CrossRef]

Saito, T.

H. Sakai, T. Saito, T. Ikeda, K. Itoh, T. Kawabata, H. Kuboki, Y. Maeda, N. Matsui, C. Rangacharyulu, M. Sasano, Y. Satou, K. Sekiguchi, K. Suda, A. Tamii, T. Uesaka, and K. Yako, “Spin correlations of strongly interacting massive fermion pairs as a test of Bell’s inequality,” Phys. Rev. Lett. 97, 150405 (2006).
[CrossRef]

Sakai, H.

H. Sakai, T. Saito, T. Ikeda, K. Itoh, T. Kawabata, H. Kuboki, Y. Maeda, N. Matsui, C. Rangacharyulu, M. Sasano, Y. Satou, K. Sekiguchi, K. Suda, A. Tamii, T. Uesaka, and K. Yako, “Spin correlations of strongly interacting massive fermion pairs as a test of Bell’s inequality,” Phys. Rev. Lett. 97, 150405 (2006).
[CrossRef]

Sanguinetti, B.

E. Pomarico, J. D. Bancal, B. Sanguinetti, A. Rochdi, and N. Gisin, “Various quantum nonlocality tests with a commercial two-photon entanglement source,” Phys. Rev. A 83, 052104 (2011).
[CrossRef]

Sank, D.

M. Ansmann, H. Wang, R. C. Bialczak, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, A. N. Cleland, and J. M. Martinis, “Violation of Bell’s inequality in Josephson phase qubits,” Nature 461, 504–506 (2009).
[CrossRef]

Sasano, M.

H. Sakai, T. Saito, T. Ikeda, K. Itoh, T. Kawabata, H. Kuboki, Y. Maeda, N. Matsui, C. Rangacharyulu, M. Sasano, Y. Satou, K. Sekiguchi, K. Suda, A. Tamii, T. Uesaka, and K. Yako, “Spin correlations of strongly interacting massive fermion pairs as a test of Bell’s inequality,” Phys. Rev. Lett. 97, 150405 (2006).
[CrossRef]

Satou, Y.

H. Sakai, T. Saito, T. Ikeda, K. Itoh, T. Kawabata, H. Kuboki, Y. Maeda, N. Matsui, C. Rangacharyulu, M. Sasano, Y. Satou, K. Sekiguchi, K. Suda, A. Tamii, T. Uesaka, and K. Yako, “Spin correlations of strongly interacting massive fermion pairs as a test of Bell’s inequality,” Phys. Rev. Lett. 97, 150405 (2006).
[CrossRef]

Sekiguchi, K.

H. Sakai, T. Saito, T. Ikeda, K. Itoh, T. Kawabata, H. Kuboki, Y. Maeda, N. Matsui, C. Rangacharyulu, M. Sasano, Y. Satou, K. Sekiguchi, K. Suda, A. Tamii, T. Uesaka, and K. Yako, “Spin correlations of strongly interacting massive fermion pairs as a test of Bell’s inequality,” Phys. Rev. Lett. 97, 150405 (2006).
[CrossRef]

Shimony, A.

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
[CrossRef]

Simon, C.

G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, “Violation of Bell’s inequality under strict Einstein locality conditions,” Phys. Rev. Lett. 81, 5039–5043(1998).
[CrossRef]

Storcz, M.

L. Wei, Y. Liu, M. Storcz, and F. Nori, “Macroscopic Einstein-Podolsky-Rosen pairs in superconducting circuits,” Phys. Rev. A 73, 052307 (2006).
[CrossRef]

Suda, K.

H. Sakai, T. Saito, T. Ikeda, K. Itoh, T. Kawabata, H. Kuboki, Y. Maeda, N. Matsui, C. Rangacharyulu, M. Sasano, Y. Satou, K. Sekiguchi, K. Suda, A. Tamii, T. Uesaka, and K. Yako, “Spin correlations of strongly interacting massive fermion pairs as a test of Bell’s inequality,” Phys. Rev. Lett. 97, 150405 (2006).
[CrossRef]

Tamii, A.

H. Sakai, T. Saito, T. Ikeda, K. Itoh, T. Kawabata, H. Kuboki, Y. Maeda, N. Matsui, C. Rangacharyulu, M. Sasano, Y. Satou, K. Sekiguchi, K. Suda, A. Tamii, T. Uesaka, and K. Yako, “Spin correlations of strongly interacting massive fermion pairs as a test of Bell’s inequality,” Phys. Rev. Lett. 97, 150405 (2006).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd ed. (Cambridge University, 1992).

Uesaka, T.

H. Sakai, T. Saito, T. Ikeda, K. Itoh, T. Kawabata, H. Kuboki, Y. Maeda, N. Matsui, C. Rangacharyulu, M. Sasano, Y. Satou, K. Sekiguchi, K. Suda, A. Tamii, T. Uesaka, and K. Yako, “Spin correlations of strongly interacting massive fermion pairs as a test of Bell’s inequality,” Phys. Rev. Lett. 97, 150405 (2006).
[CrossRef]

Vallone, G.

G. Lima, G. Vallone, A. Chiuri, A. Cabello, and P. Mataloni, “Experimental Bell-inequality violation without the postselection loophole,” Phys. Rev. A 81, 040101(R) (2010).
[CrossRef]

Verstraete, F.

F. Verstraete and M. M. Wolf, “Entanglement versus Bell violations and their behavior under local filtering operations,” Phys. Rev. Lett. 89, 170401 (2002).
[CrossRef]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd ed. (Cambridge University, 1992).

Walls, D. F.

D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, 1994).

Wang, H.

M. Ansmann, H. Wang, R. C. Bialczak, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, A. N. Cleland, and J. M. Martinis, “Violation of Bell’s inequality in Josephson phase qubits,” Nature 461, 504–506 (2009).
[CrossRef]

Wei, L.

L. Wei, Y. Liu, M. Storcz, and F. Nori, “Macroscopic Einstein-Podolsky-Rosen pairs in superconducting circuits,” Phys. Rev. A 73, 052307 (2006).
[CrossRef]

L. Wei, Y. Liu, and F. Nori, “Testing Bells inequality in a constantly coupled Josephson circuit by effective single-qubit operations,” Phys. Rev. B 72, 104516 (2005).
[CrossRef]

Weides, M.

M. Ansmann, H. Wang, R. C. Bialczak, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, A. N. Cleland, and J. M. Martinis, “Violation of Bell’s inequality in Josephson phase qubits,” Nature 461, 504–506 (2009).
[CrossRef]

Weihs, G.

G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, “Violation of Bell’s inequality under strict Einstein locality conditions,” Phys. Rev. Lett. 81, 5039–5043(1998).
[CrossRef]

Weinfurter, H.

G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, “Violation of Bell’s inequality under strict Einstein locality conditions,” Phys. Rev. Lett. 81, 5039–5043(1998).
[CrossRef]

P. G. Kwiat, K. Mattle, H. Weinfurter, and A. Zeilinger, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[CrossRef]

Wenner, J.

M. Ansmann, H. Wang, R. C. Bialczak, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, A. N. Cleland, and J. M. Martinis, “Violation of Bell’s inequality in Josephson phase qubits,” Nature 461, 504–506 (2009).
[CrossRef]

Werner, R. F.

R. F. Werner, “Quantum states with Einstein Podolsky Rosen correlations admitting a hidden-variable model,” Phys. Rev. A 40, 4277–4281 (1989).
[CrossRef]

White, A. G.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[CrossRef]

Wineland, D. J.

M. A. Rowe, D. Kielpinski, V. Meyer, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland, “Experimental violation of a Bell’s inequality with efficient detection,” Nature 409, 791–794 (2001).
[CrossRef]

Wolf, M. M.

F. Verstraete and M. M. Wolf, “Entanglement versus Bell violations and their behavior under local filtering operations,” Phys. Rev. Lett. 89, 170401 (2002).
[CrossRef]

Wootters, W. K.

V. Coffman, J. Kundu, and W. K. Wootters, “Distributed entanglement,” Phys. Rev. A 61, 052306 (2000).
[CrossRef]

Yako, K.

H. Sakai, T. Saito, T. Ikeda, K. Itoh, T. Kawabata, H. Kuboki, Y. Maeda, N. Matsui, C. Rangacharyulu, M. Sasano, Y. Satou, K. Sekiguchi, K. Suda, A. Tamii, T. Uesaka, and K. Yako, “Spin correlations of strongly interacting massive fermion pairs as a test of Bell’s inequality,” Phys. Rev. Lett. 97, 150405 (2006).
[CrossRef]

Zeilinger, A.

G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, “Violation of Bell’s inequality under strict Einstein locality conditions,” Phys. Rev. Lett. 81, 5039–5043(1998).
[CrossRef]

P. G. Kwiat, K. Mattle, H. Weinfurter, and A. Zeilinger, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[CrossRef]

Zhang, Y. S.

Y. F. Huang, C. F. Li, Y. S. Zhang, and G. C. Guo, “Experimental test of CHSH inequality for non-maximally entangled states,” Phys. Lett. Sect. A 287, 317–320 (2001).
[CrossRef]

Am. J. Phys. (1)

D. Dehlinger and M. W. Mitchell, “Entangled photons, nonlocality, and Bell inequalities in the undergraduate laboratory,” Am. J. Phys. 70, 903–910 (2002).
[CrossRef]

Lett. Math. Phys. (1)

B. S. Cirel’son, “Quantum generalizations of Bell’s inequality,” Lett. Math. Phys. 4, 93–100 (1980).
[CrossRef]

Nature (2)

M. A. Rowe, D. Kielpinski, V. Meyer, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland, “Experimental violation of a Bell’s inequality with efficient detection,” Nature 409, 791–794 (2001).
[CrossRef]

M. Ansmann, H. Wang, R. C. Bialczak, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, A. N. Cleland, and J. M. Martinis, “Violation of Bell’s inequality in Josephson phase qubits,” Nature 461, 504–506 (2009).
[CrossRef]

Phys. Lett. A (1)

R. Horodecki, P. Horodecki, and M. Horodecki, “Violating Bell inequality by mixed spin-1/2 states: necessary and sufficient condition,” Phys. Lett. A 200, 340–344 (1995).
[CrossRef]

Phys. Lett. Sect. A (1)

Y. F. Huang, C. F. Li, Y. S. Zhang, and G. C. Guo, “Experimental test of CHSH inequality for non-maximally entangled states,” Phys. Lett. Sect. A 287, 317–320 (2001).
[CrossRef]

Phys. Rev. (1)

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?,” Phys. Rev. 47, 777–780 (1935).
[CrossRef]

Phys. Rev. A (6)

E. Pomarico, J. D. Bancal, B. Sanguinetti, A. Rochdi, and N. Gisin, “Various quantum nonlocality tests with a commercial two-photon entanglement source,” Phys. Rev. A 83, 052104 (2011).
[CrossRef]

R. F. Werner, “Quantum states with Einstein Podolsky Rosen correlations admitting a hidden-variable model,” Phys. Rev. A 40, 4277–4281 (1989).
[CrossRef]

G. Lima, G. Vallone, A. Chiuri, A. Cabello, and P. Mataloni, “Experimental Bell-inequality violation without the postselection loophole,” Phys. Rev. A 81, 040101(R) (2010).
[CrossRef]

L. Wei, Y. Liu, M. Storcz, and F. Nori, “Macroscopic Einstein-Podolsky-Rosen pairs in superconducting circuits,” Phys. Rev. A 73, 052307 (2006).
[CrossRef]

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[CrossRef]

V. Coffman, J. Kundu, and W. K. Wootters, “Distributed entanglement,” Phys. Rev. A 61, 052306 (2000).
[CrossRef]

Phys. Rev. B (1)

L. Wei, Y. Liu, and F. Nori, “Testing Bells inequality in a constantly coupled Josephson circuit by effective single-qubit operations,” Phys. Rev. B 72, 104516 (2005).
[CrossRef]

Phys. Rev. Lett. (8)

H. Sakai, T. Saito, T. Ikeda, K. Itoh, T. Kawabata, H. Kuboki, Y. Maeda, N. Matsui, C. Rangacharyulu, M. Sasano, Y. Satou, K. Sekiguchi, K. Suda, A. Tamii, T. Uesaka, and K. Yako, “Spin correlations of strongly interacting massive fermion pairs as a test of Bell’s inequality,” Phys. Rev. Lett. 97, 150405 (2006).
[CrossRef]

P. G. Kwiat, K. Mattle, H. Weinfurter, and A. Zeilinger, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[CrossRef]

G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, “Violation of Bell’s inequality under strict Einstein locality conditions,” Phys. Rev. Lett. 81, 5039–5043(1998).
[CrossRef]

S. L. Braunstein, A. Mann, and M. Revzen, “Maximal violation of Bell inequalities for mixed states,” Phys. Rev. Lett. 68, 3259–3261 (1992).
[CrossRef]

D. Magde and H. Mahr, “Study in ammonium dihydrogen phosphate of spontaneous parametric interaction tunable from 4400 to 16 000 A,” Phys. Rev. Lett. 18, 905–907(1967).
[CrossRef]

F. Verstraete and M. M. Wolf, “Entanglement versus Bell violations and their behavior under local filtering operations,” Phys. Rev. Lett. 89, 170401 (2002).
[CrossRef]

A. Aspect, P. Grangier, and G. Roger, “Experimental tests of realistic local theories via Bell’s theorem,” Phys. Rev. Lett. 47, 460–463 (1981).
[CrossRef]

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
[CrossRef]

Physics (1)

J. Bell, “On the Einstein Podolsky Rosen paradox,” Physics 1, 195–200 (1964).

Other (5)

D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, 1994).

Http://www.quantum-info.com/ .

P. L. Knight and A. Miller, Measuring the Quantum States of Light (Cambridge University, 1997).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd ed. (Cambridge University, 1992).

M. A. Nielsen and I. L. Chuang, Quantum Computation and Information (Cambridge University, 2000).

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Figures (2)

Fig. 1.
Fig. 1.

Experimental scheme. s and i, signal and idler photons; λ/4 and λ/2, quarter- and half-wave plates; PBS, polarization beam splitter; C, fiber coupler with long-pass filter; F, single-mode fiber; APD, avalanche single photon detector; Co., coincidence-counting modular. The dash-dotted box represents the mixed polarization entangled-photon source, and the dotted box is the additional devices inserted in the commercial system.

Fig. 2.
Fig. 2.

Various projection measurements and their optical realizations.

Tables (3)

Tables Icon

Table 1. Coincidence Counts for the Selected 16 Sets of Linearly Independent Projection Measurements in a Time Interval of 1 s

Tables Icon

Table 2. Coincidence Counting Number in a Time Interval of 1 s Under the Measurement Setting Op

Tables Icon

Table 3. Coincidence Counts in a Time Interval of 1 s Under the Optimized Measurement Setting Ot

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

|ψp=12(|Hs|Vi+|Vs|Hi),
ρ^=ν=116Γ^νrν.
nν=Nψν|ρ^|ψν,
ρ^=N1ν=116M^νnν,M^ν=ν=116(B1)ν,μΓ^μ,
ρ^t=(0.01860.0030+0.0648i0.04360.0193i0.01420.0393i0.00300.0648i0.46940.27820.0485i0.0258+0.0272i0.0436+0.0193i0.2782+0.0485i0.49970.0386+0.0176i0.0142+0.0393i0.02580.0272i0.03860.0176i0.0123).
Tr(ρ^t2)=0.653<1,
C=max{0,r1r2r3r4}=0.479<1.
Smax=21+C2=2.218.
S2,
S=|E(θs,θi)+E(θs,θi)E(θs,θi)+E(θs,θi)|.
E(θs,θi)=P(θs,θi)+P(θs+π2,θi+π2)P(θs,θi+π2)P(θs+π2,θi),
|ψν=cos(θs)cos(θi)|Hs|Hi+cos(θs)sin(θi)|Hs|Vi+sin(θs)cos(θi)|Vs|Hi+sin(θs)sin(θi)|Vs|Vi,
Sp=2.114±0.015,
E(θs,θi)=n(θs,θi)+n(θs+π/2,θi+π/2)n(θs,θi+π/2)n(θs+π/2,θi)n(θs,θi)+n(θs+π/2,θi+π/2)+n(θs,θi+π/2)+n(θs+π/2,θi),
St=2.293±0.014.
σS=ν=116σnν2(Snν)2=ν=116nν(Snν)2.

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